RETURN TO EQUILIBRIUM V.Jak si c, C. A. Pillet, J. Derezi nski. - - PowerPoint PPT Presentation

RETURN TO EQUILIBRIUM V.Jakˇ si´ c,

• C. A. Pillet,
• J. Derezi´

nski. Conventional wisdom (1) In a generic situation, a small system interacting with a large reservoir at temperature T goes to equi- librium at the same temperature. (2) The behavior of a small system interacting with reservoirs at distinct temperatures is much more dif- ficult to desribe than in the case of a reservoir at a fixed temperature.

Rigorous expression

• f conventional wisdom

Rigorous theorems proven for nontrivial, explicit and realistic models: (1) Return to equilibrium for a generic small system interacting with a thermal reservoir. (2) Absence of normal stationary states for a generic small system interacting with a non-equilibrium reser- voir.

Mathematical techniques involved in the study of return to equilibrium.

• Operator algebras:

– KMS states, – Standard forms, Liouvilleans;

• Quantum field theory:

– Quasi-free (Araki-Woods) representations of the CCR;

• Spectral theory:

– Fermi Golden Rule, the Feshbach method, – The positive commutator (Mourre) method.

Plan of the lecture (1) Small system interacting with a bosonic reservoir (2) W ∗-algebraic background. (3) Fermi golden rule

SMALL SYSTEM INTERACTING WITH A BOSONIC RESERVOIR Bosonic Fock space Γs(L2(Rd)) :=

∞

⊕

n=0 ⊗n s L2(Rd).

Creation/annihilation operators: [a∗(ξ1), a∗(ξ2)] = 0, [a(ξ1), a(ξ2)] = 0, [a∗(ξ1), a(ξ2)] = δ(ξ1 − ξ2).

• a∗(ξ)f(ξ)dξΦ :=

√ n + 1f ⊗s Φ, Φ ∈ ⊗n

s L2(Rd).

Vacuum: Ω = 1 ∈ ⊗0

sL2(Rd) = C.

Free Hamiltonian of fotons or phonons: H =

• |ξ|a∗(ξ)a(ξ)dξ.

Quasi-free representations of the CCR Radiation density: Rd ∋ ξ → ρ(ξ) ∈ [0, ∞[. We look for creation/annihillation operators a∗

ρ,l(ξ)/aρ,l(ξ), with

a quasi-free state of density ρ given by a cyclic vector Ω: [a∗

ρ,l(ξ1), a∗ ρ,l(ξ2)] = 0,

[aρ,l(ξ1), aρ,l(ξ2)] = 0, [aρ,l(ξ1), a∗

ρ,l(ξ2)] = δ(ξ1 − ξ2).

Wρ,l(f) := exp i √ 2

• (f(ξ)a∗

ρ,l(ξ) + f(ξ)aρ,l(ξ))dξ

• .

(Ω|Wρ,l(f)Ω) = exp

• −1

4

• |f(ξ)|2(1 + 2ρ(ξ))dξ
• .

Araki-Woods representation of CCR We will write a∗

l (ξ)/al(ξ), ar(ξ)/a∗ r(ξ) for the creation/

annihilation operators corresponding to the left and right L2(Rd) resp. acting on the Fock space Γs(L2(Rd) ⊕ L2(Rd)). Left Araki-Woods creation/annihillation operators are defined as a∗

ρ,l(ξ) :=

• 1 + ρ(ξ)a∗

l (ξ) +

• ρ(ξ)ar(ξ),

aρ,l(ξ) :=

• 1 + ρ(ξ)al(ξ) +
• ρ(ξ)a∗

r(ξ).

Left Araki-Woods algebra is denoted by MAW

ρ,l

and de- fined as the W ∗-algebra generated by the operators Wρ,l(f). The vacuum Ω defines a quasi-free state of density ρ.

Commutant of the Araki-Woods algebra Define an involution ǫ on L2(Rd) ⊕ L2(Rd) by ǫ(f1, f2) := (f1, f2). Set J := Γ(ǫ). Then J is the modular involution for the state (Ω| · Ω). Right Araki-Woods creation/annihillation operators: a∗

ρ,r(ξ) :=

• ρ(ξ)al(ξ) +
• 1 + ρ(ξ)a∗

r(ξ),

aρ,r(ξ) :=

• ρ(ξ)a∗

l (ξ) +

• 1 + ρ(ξ)ar(ξ).

generate the right Araki-Woods algebra denoted by MAW

ρ,r . Note that JMAW ρ,l J = MAW ρ,r is the commutant of

MAW

ρ,l .

Dynamics of the quasifree bosons The Liouvillean of free bosons: L =

• |ξ|a∗

l (ξ)al(ξ)dξ −

• |ξ|a∗

r(ξ)ar(ξ)dξ.

Note that JLJ = −L. eitL · e−itL defines a dynamics on MAW

ρ,l . The state (Ω|·Ω)

is β-KMS iff the density is given by the Planck law: ρ(ξ) = (eβ|ξ| −1)−1.

Small quantum system in contact with Bose gas at zero density Hilbert space of the small quantum system: K = Cn. The Hamiltonian of the free system: K. The free Pauli-Fierz Hamiltonian: Hfr := K ⊗ 1 + 1 ⊗

• |ξ|a∗(ξ)a(ξ)dξ.

Rd ∋ ξ → v(ξ) ∈ B(K) describes the interaction: V :=

• v(ξ) ⊗ a∗(ξ)dξ + hc

The full Pauli-Fierz Hamiltonian: H := Hfr + λV. The Pauli-Fierz system at zero density:

• B(K ⊗ Γs(L2(Rd)), eitH · e−itH

.

Small quantum system in contact with Bose gas at density ρ. The algebra of observables of the composite system: Mρ := B(K) ⊗ Mρ,l ⊂ B

• K ⊗ Γs(L2(Rd) ⊕ L2(Rd))
• .

The free Pauli-Fierz semi-Liouvillean at density ρ: Lsemi

fr

:= K ⊗ 1 + 1 ⊗

• |ξ|a∗

l (ξ)al(ξ)dξ −

• |ξ|a∗

r (ξ)ar(ξ)dξ

• .

The interaction: Vρ :=

• v(ξ) ⊗ a∗

ρ,l(ξ)dξ + hc.

The full Pauli-Fierz semi-Liouvillean at density ρ: Lsemi

ρ

:= Lsemi

fr

+ λVρ. The Pauli-Fierz W ∗-dynamical system at density ρ: (Mρ, σρ), where σρ,t(A) := eitLsemi

ρ

A e−itLsemi

ρ

.

Relationship between the dynamics at zero density and at density ρ. Set ρ = 0. M0 ≃ B(K ⊗ Γs(L2(Rd)) ⊗ 1. Lsemi ≃ H ⊗ 1 − 1 ⊗

• |ξ|ar(∗(ξ)ar(ξ)dξ.

σ0,t(A ⊗ 1) = eitH A e−itH ⊗1. If we formally replace al(ξ), ar(ξ) with aρ,l(ξ), aρ,r(ξ) (the CCR do not change!) then M0, Lsemi , σ0 trans- form into Mρ, Lsemi

ρ

, σρ. In the case of a finite num- ber of degrees of freedom this can be implemented by a unitary Bogoliubov transformation. (Mρ, σρ) can be viewed as a thermodynamical limit of (M0, σ0).

Theorem I: Return equilibrium in the thermal case. Let the reservoir have inverse temperature β. Assume some conditions about the regularity and effective- ness of v(ξ). Then there exists λ0 > 0 such that for 0 < |λ| ≤ λ0, (Mρ, σρ) has a single normal stationary state ω. This state is β-KMS and for any normal state φ and A ∈ Mρ, we have lim|t|→∞ φ(σρ,t(A)) = ω(A). Jakˇ si´ c- Pillet, Jakˇ si´ c-D., Bach-Fr¨

• hlich-Sigal, Fr¨
• hlich-Merkli

Theorem II: Absence of normal stationary states in the non-equilibrium case. Suppose that the reservoir has parts at distinct temperatures. Assume some con- ditions about the regularity and effectiveness of v(ξ). Then there exists λ0 > 0 such that for 0 < |λ| ≤ λ0, (Mρ, σρ) has no normal stationary states. Jakˇ si´ c-D.

Standard representation of Mρ. In order to prove the above theorems we need to go to the standard representation: π : Mρ → B(K ⊗ K ⊗ Γs(L2(Rd) ⊕ L2(Rd)), π(A ⊗ B) = A ⊗ 1 ⊗ B, JΦ1 ⊗ Φ2 ⊗ Ψ = Φ2 ⊗ Φ1 ⊗ Γ(ǫ)Ψ. The free Pauli-Fierz Liouvillean: Lfr := K ⊗ 1 ⊗ 1 − 1 ⊗ K ⊗ 1 +1 ⊗ 1 ⊗ |ξ|

• a∗

l (ξ)al(ξ) − a∗ l (ξ)al(ξ)

• dξ,

π(Vρ) =

• v(ξ) ⊗ 1 ⊗ a∗

ρ,l(ξ)dξ + hc,

Jπ(Vρ)J =

• 1 ⊗ v(ξ) ⊗ 1 ⊗ a∗

ρ,r(ξ)dξ + hc.

The full Pauli-Fierz Liouvillean at density ρ: Lρ = Lfr + λπ(Vρ) − λJπ(Vρ)J.

Theorem I’: Let the reservoir have inverse tempera- ture β. Assume some conditions about the regularity and effectiveness of v(ξ). Then there exists λ0 > 0 such that for 0 < |λ| ≤ λ0, dim KerLρ = 1 and Lρ has abso- lutely continuous spectrum away from 0. Theorem II’: Absence of normal stationary states in the non-equilibrium case. Suppose that the reservoir has parts at distinct temperatures. Assume some con- ditions about the regularity and effectiveness of v(ξ). Then there exists λ0 > 0 such that for 0 < |λ| ≤ λ0, dim KerLρ = 0.

Spectrum of Pauli-Fierz Liouvillean Spectrum of Lfr is R. Point spectrum of Lfr is spK − spK. Φfr := e−βK/2 ⊗Ω is a β-KMS vector of Lfr. By Araki-Jakˇ si´ c-Pillet-D, e−(Lfr+λπ(Vρ))β/2 Φfr is a β-KMS vector of Lρ. Therefore, KerLρ ≥ 1. By a rigorous version of the Fermi Golden Rule, if the interaction is sufficiently regular and effective, then there exists λ0 > 0 such that for 0 < |λ| ≤ λ0 KerLρ ≤ 1.

W ∗-ALGEBRAIC BACKGROUND 2 approaches to quantum systems (1) C∗-dynamical system (A, αt): A – C∗-algebra, t → αt ∈ Aut(A) – strongly contin- uous 1-parameter group. (2) W ∗-dynamical system (M, σt): M – W ∗-algebra, t → σt ∈ Aut(M) – σ-weakly con- tinuous 1-parameter group. We use the W ∗-dynamical approach

The GNS representation Suppose that ω is a state on M. Then we have the GNS representation π : M → B(H) with Ω ∈ H – a cyclic vector for π(M) such that ω(A) = (Ω|π(A)Ω), A ∈ M. If ω is normal, then so is π. If in addition ω is stationary wrt a W ∗-dynamics σ, then we have a distinguished unitary implementation

• f σ:

π(σt(A)) = eitL π(A) e−itL, A ∈ M, LΩ = 0.

Theorem Return to equilibrium in mean. Suppose that ω is faithful. Then the following state- ments are equivalent: (1) ω is a unique invariant normal state. (2) Ω is a unique eigenvector of L. (3) For any normal state φ and A ∈ M, lim

t→∞

1 t t φ(σs(A))ds = ω(A). Theorem Return to equilibrium. Suppose that ω is faithful and L has absolutely con- tinuous spectrum away from 0. Then for any normal state φ and A ∈ M, lim

t→∞ φ(σt(A)) = ω(A).

Standard form of a W ∗-algebra Connes, Araki, Haagerup A W ∗-algebra in a standard form is a quadruple (M, H, J, H+), where H is a Hilbert space, M ⊂ B(H) is a W ∗-algebra, J is an antiunitary involution on H (that is, J is anti- linear, J2 = 1, J∗ = J) and H+ is a self-dual cone in H such that: (1) JMJ = M′; (2) JAJ = A∗ for A in the center of M; (3) JΨ = Ψ for Ψ ∈ H+; (4) AJAH+ ⊂ H+ for A ∈ M.

Standard form in a GNS representation If ω is a faithful state, (π, H, Ω) – the corresponding GNS representation, J – the modular conjugation given by the Tomita- Takesaki theory, H+ := {π(A)Jπ(A)Ω : A ∈ M}cl, then (π(M), H, JΩ, H+), is a standard form.

Standard Liouvillean For every W ∗-dynamics σ there exists a unique self- adjoint operator L called the Liouvillean of σ such that π(σt(A)) = eitL π(A) e−itL, A ∈ M, eitL H+ ⊂ H+. If the W ∗-dynamics σ has a faithful invariant normal state ω, then its Liouvilean L coincides with the oper- ator L introduced in the GNS representation.

Normal states and vectors in the positive cone Theorem Every normal state ω has a unique standard vector representative, that is a vector Ω ∈ H+ such that ω(A) = (Ω|π(A)Ω), A ∈ M. Theorem (1) dim KerL = 0 iff the W ∗-dynamics σt has no normal invariant states. (2) dim KerL = 1 iff the W ∗-dynamics σt has a single normal invariant state.

KMS states Let σ be a W ∗-dynamics and L the corresponding Li-

• uvillean.

A normal state ω is called a β-KMS state iff ω(AB) = ω(Bσiβ(A)), A, B ∈ M, B σ − analytic. β-KMS states are stationary. A vector Ω is called a β-KMS vector iff Ω ∈ H+ and e−βL/2 AΩ = JA∗Ω, A ∈ M. β-KMS vectors belong to KerL. They are standard vector representatives of β-KMS states.

Theorem Let (M, σfr) be a W ∗-dynamical system with the Liouvillean Lfr. Let Ωfr be a β-KMS vector for σfr. Let V be a self-adjoint operator affiliated to M satisfying some technical assumptions. Then (1) There exists a perturbed dynamics σ such that d dtσt(A) = d dtσfr,t(A) + i[V, σfr,t(A)]. (2) The Liouvillean of σ equals L = Lfr + π(V ) − Jπ(V )J. (3) e−βπ(V )/2 Ω is a β-KMS vector for σ. Araki–bounded V ; Jakˇ si´ c, Pillet and D.–unbounded V .

Example: type I factor W ∗-algebra: B(K); Standard Hilbert space: K⊗K = B2(K); Standard representation: π(A) = A ⊗ 1K ≃ A ·; Standard positive cone: B2

+(K);

State: ω(A) = TrρA, ρ ∈ B1

+(K), Trρ = 1;

Its vector representative: Ω = ρ1/2 ∈ B2

+(K);

W ∗-dynamics: σt(A) = eitK A e−itK; Its Liouvillean: L = K ⊗ 1 − 1 ⊗ K ≃ [K, ·]; β-KMS state: ωβ(A) = (Tr e−βK)−1Tr eβK A; β-KMS vector: (Tr e−βK)−1/2 eβK/2. —————————————————-

B2(K)–Hilbert-Schmidt operators, B1(K)–trace class operators

RIGOROUS FERMI GOLDEN RULE 2nd order perturbation theory for isolated eigenvalues Unperturbed operator: L0. The spectral projection

• nto an isolated part of spectrum of L0 consisting of a

finite number of eigenvalues is denoted P. We define L0P =: E =

• e∈spE

e1e(E). Perturbation: Q. We assume that there is no 1st order shift of eigenvalues: PQP = 0. Perturbed operator: Lλ := L0 + λQ.

• Theorem. For small λ, in a neighborhood of spE we

have spLλ = sp(E + λ2Γ) + o(λ2), where Γ is the Level Shift Operator Γ =

• e∈spE

1e(E)Q(e − L0)−1Q1e(E). Multiplicities of eigenvalues of E + λ2Γ coincide with multiplicities of corresponding clusters of eigenvalues

• f Lλ.

2nd order perturbation theory for isolated eigenvalues Let L0, P, Q and Lλ be as above, except that the spec- trum of E can be embedded in the rest of spectrum of

• L0. Introduce the (upper) Level Shift Operator:

Γ =

• e∈spE

lim

ǫ↓0 1e(E)Q(e + iǫ − L0)−1Q1e(E).

Clearly, Γ satisfies ΓE = EΓ, 1 2i(Γ − Γ∗) ≤ 0. Fermi Golden Rule:

1 2(Γ + Γ∗) describes energy shift, 1 2i(Γ − Γ∗) describes the decay rates.

• Theorem. There exists λ0 > 0 such that for 0 < |λ| < λ0

dim 1p(Lλ) ≤ dim Ker 1 2i(Γ − Γ∗). Proofs (for Pauli-Fierz Liouvilleans) involve 1) analytic deformation method, Jakˇ si´ c-Pillet; 2) positive commutator method, Jakˇ si´ c-D; Merkli; 3) “renormalization group” Bach-Fr¨

• hlich-Sigal.

2nd order perturbation theory applied to Pauli-Fierz Liouvilleans Unperturbed operator: L0. Projection: 1p(Lfr), which coincides with the projec- tion onto K ⊗ K ⊗ Ω. Perturbation: π(Vρ) − Jπ(Vρ)J. Perturbed operator: Lρ.

• Theorem. If the interaction is sufficiently regular and

effective, then dim Ker 1

2i(Γ − Γ∗) ≤ 1 in thermal case,

dim Ker 1

2i(Γ − Γ∗) = 0 in nonthermal case.